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## Due November 13

### Problem 1 (Bessel inequality)

Let $X$ be a separable Hilbert space and $\{v_k\}$ an orthonormal set (may be finite). Then

$\displaystyle \sum_k |(x,v_k)|^2 \le ||x||^2.$

### Problem 2

Consider the function $||\cdot||_\infty:l^2\to[0,\infty)$ given by

$||(a_n)||_\infty = \sup\{|a_n|:n\ge 1\}$.

1. $||\cdot||_\infty$ is a norm in $l^2$

2. Is $||\cdot||_\infty$ equivalent to the $l^2$-norm?

3. Is $(l^2,||\cdot||_\infty)$ complete?

### Problem 3

Let $X$ be a separable Hilbert space and $Y$ a closed subspace. Then $Y$ is separable.

### Problem 4

Let $X$ be a separable inner product space and $\bar X$ its completion. Then $\bar X$ is a separable Hilbert space.

### Problem 5

Let $\phi\in X'$, where $X$ is a Hilbert space. Then $\ker\phi$ is a closed subspace of $X$ of codimension 1.